The Liar’s paradox is one of the oldest and most profound paradoxes known. It is attributed to the Greek philosopher Eubulides of Miletus, who lived in the 4th century BC. The Liar’s paradox can be expressed very simply, as it arises directly from the consideration of the following statement: “This sentence is false”. The paradox arises from the fact that if the latter sentence is true, then it follows that it is false; but if the same sentence is false, then it is false that it is false and therefore true. Thus “This sentence is false” is false if it is true, and true if it is false. In conclusion, “This sentence is false” is true if and only if it is false. And this last conclusion is paradoxical.
We often denote “This sentence is false” by (λ). It is useful at this point, to describe in detail the different steps of reasoning that lead to the Liar’s paradox (the symbol ∴ denotes the conclusion here):
(λ) (λ) is false
(1) (λ) is either true or false [bivalence]
(2) if (λ) is true hypothesis 1
(3) then it is true that (λ) is false [of (λ),(2)]
(4) then (λ) is false [from (3)
(5) if (λ) is false assumption 2
(6) then it is false that (λ) is false from (λ),(5)
(7) then (λ) is true [from (6)
(8) ∴ (λ) is neither true nor false [from (4),(7)]
The conclusion (8) here is paradoxical, since it follows that (λ) is neither true nor false, in contradiction with the principle (1) of bivalence. The problem raised by the Liar is thus the following: what is the truth value of the proposition (λ), given that it cannot be assigned, without contradiction, the truth value true or false?
(excerpt from) Franceschi P. An Introduction to Analytic Philosophy: Paradoxes, Arguments and Contemporary Problems, 2nd edition, March, 2010